Step |
Hyp |
Ref |
Expression |
1 |
|
imo72b2.1 |
|- ( ph -> F : RR --> RR ) |
2 |
|
imo72b2.2 |
|- ( ph -> G : RR --> RR ) |
3 |
|
imo72b2.4 |
|- ( ph -> B e. RR ) |
4 |
|
imo72b2.5 |
|- ( ph -> A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) ) |
5 |
|
imo72b2.6 |
|- ( ph -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 ) |
6 |
|
imo72b2.7 |
|- ( ph -> E. x e. RR ( F ` x ) =/= 0 ) |
7 |
2 3
|
ffvelrnd |
|- ( ph -> ( G ` B ) e. RR ) |
8 |
7
|
recnd |
|- ( ph -> ( G ` B ) e. CC ) |
9 |
8
|
abscld |
|- ( ph -> ( abs ` ( G ` B ) ) e. RR ) |
10 |
|
1red |
|- ( ph -> 1 e. RR ) |
11 |
|
simpr |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 1 < ( abs ` ( G ` B ) ) ) |
12 |
2
|
adantr |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> G : RR --> RR ) |
13 |
3
|
adantr |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> B e. RR ) |
14 |
12 13
|
ffvelrnd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( G ` B ) e. RR ) |
15 |
14
|
recnd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( G ` B ) e. CC ) |
16 |
15
|
abscld |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) e. RR ) |
17 |
10
|
adantr |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 1 e. RR ) |
18 |
|
ax-resscn |
|- RR C_ CC |
19 |
|
imaco |
|- ( ( abs o. F ) " RR ) = ( abs " ( F " RR ) ) |
20 |
19
|
eqcomi |
|- ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR ) |
21 |
|
imassrn |
|- ( ( abs o. F ) " RR ) C_ ran ( abs o. F ) |
22 |
21
|
a1i |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( ( abs o. F ) " RR ) C_ ran ( abs o. F ) ) |
23 |
1
|
adantr |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> F : RR --> RR ) |
24 |
|
absf |
|- abs : CC --> RR |
25 |
24
|
a1i |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> abs : CC --> RR ) |
26 |
18
|
a1i |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> RR C_ CC ) |
27 |
25 26
|
fssresd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs |` RR ) : RR --> RR ) |
28 |
23 27
|
fco2d |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs o. F ) : RR --> RR ) |
29 |
28
|
frnd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ran ( abs o. F ) C_ RR ) |
30 |
22 29
|
sstrd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( ( abs o. F ) " RR ) C_ RR ) |
31 |
20 30
|
eqsstrid |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs " ( F " RR ) ) C_ RR ) |
32 |
|
0re |
|- 0 e. RR |
33 |
32
|
ne0ii |
|- RR =/= (/) |
34 |
33
|
a1i |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> RR =/= (/) ) |
35 |
34 28
|
wnefimgd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( ( abs o. F ) " RR ) =/= (/) ) |
36 |
35
|
necomd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> (/) =/= ( ( abs o. F ) " RR ) ) |
37 |
20
|
a1i |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR ) ) |
38 |
36 37
|
neeqtrrd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> (/) =/= ( abs " ( F " RR ) ) ) |
39 |
38
|
necomd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs " ( F " RR ) ) =/= (/) ) |
40 |
|
simpr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ c = 1 ) -> c = 1 ) |
41 |
40
|
breq2d |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ c = 1 ) -> ( t <_ c <-> t <_ 1 ) ) |
42 |
41
|
ralbidv |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ c = 1 ) -> ( A. t e. ( abs " ( F " RR ) ) t <_ c <-> A. t e. ( abs " ( F " RR ) ) t <_ 1 ) ) |
43 |
1 5
|
extoimad |
|- ( ph -> A. t e. ( abs " ( F " RR ) ) t <_ 1 ) |
44 |
43
|
adantr |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> A. t e. ( abs " ( F " RR ) ) t <_ 1 ) |
45 |
17 42 44
|
rspcedvd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> E. c e. RR A. t e. ( abs " ( F " RR ) ) t <_ c ) |
46 |
31 39 45
|
suprcld |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. RR ) |
47 |
18 46
|
sselid |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. CC ) |
48 |
18 16
|
sselid |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) e. CC ) |
49 |
47 48
|
mulcomd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) x. ( abs ` ( G ` B ) ) ) = ( ( abs ` ( G ` B ) ) x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) ) |
50 |
32
|
a1i |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 0 e. RR ) |
51 |
|
0lt1 |
|- 0 < 1 |
52 |
51
|
a1i |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 0 < 1 ) |
53 |
50 17 16 52 11
|
lttrd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 0 < ( abs ` ( G ` B ) ) ) |
54 |
53
|
gt0ne0d |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) =/= 0 ) |
55 |
46 16 54
|
redivcld |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) / ( abs ` ( G ` B ) ) ) e. RR ) |
56 |
23
|
adantr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> F : RR --> RR ) |
57 |
12
|
adantr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> G : RR --> RR ) |
58 |
|
simpr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> u e. RR ) |
59 |
13
|
adantr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> B e. RR ) |
60 |
|
simpr |
|- ( ( ph /\ v = B ) -> v = B ) |
61 |
60
|
oveq2d |
|- ( ( ph /\ v = B ) -> ( u + v ) = ( u + B ) ) |
62 |
61
|
fveq2d |
|- ( ( ph /\ v = B ) -> ( F ` ( u + v ) ) = ( F ` ( u + B ) ) ) |
63 |
60
|
oveq2d |
|- ( ( ph /\ v = B ) -> ( u - v ) = ( u - B ) ) |
64 |
63
|
fveq2d |
|- ( ( ph /\ v = B ) -> ( F ` ( u - v ) ) = ( F ` ( u - B ) ) ) |
65 |
62 64
|
oveq12d |
|- ( ( ph /\ v = B ) -> ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) ) |
66 |
60
|
fveq2d |
|- ( ( ph /\ v = B ) -> ( G ` v ) = ( G ` B ) ) |
67 |
66
|
oveq2d |
|- ( ( ph /\ v = B ) -> ( ( F ` u ) x. ( G ` v ) ) = ( ( F ` u ) x. ( G ` B ) ) ) |
68 |
67
|
oveq2d |
|- ( ( ph /\ v = B ) -> ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) ) |
69 |
65 68
|
eqeq12d |
|- ( ( ph /\ v = B ) -> ( ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) <-> ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) ) ) |
70 |
69
|
ralbidv |
|- ( ( ph /\ v = B ) -> ( A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) <-> A. u e. RR ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) ) ) |
71 |
|
ralcom |
|- ( A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) <-> A. v e. RR A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) ) |
72 |
71
|
biimpi |
|- ( A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) -> A. v e. RR A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) ) |
73 |
72
|
a1i |
|- ( ph -> ( A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) -> A. v e. RR A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) ) ) |
74 |
73
|
imp |
|- ( ( ph /\ A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) ) -> A. v e. RR A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) ) |
75 |
4 74
|
mpdan |
|- ( ph -> A. v e. RR A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) ) |
76 |
70 3 75
|
rspcdv2 |
|- ( ph -> A. u e. RR ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) ) |
77 |
76
|
r19.21bi |
|- ( ( ph /\ u e. RR ) -> ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) ) |
78 |
77
|
adantlr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) ) |
79 |
5
|
ad2antrr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 ) |
80 |
56 57 58 59 78 79
|
imo72b2lem0 |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( ( abs ` ( F ` u ) ) x. ( abs ` ( G ` B ) ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) ) |
81 |
|
0xr |
|- 0 e. RR* |
82 |
81
|
a1i |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 0 e. RR* ) |
83 |
|
1xr |
|- 1 e. RR* |
84 |
83
|
a1i |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 1 e. RR* ) |
85 |
16
|
adantr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( abs ` ( G ` B ) ) e. RR ) |
86 |
85
|
rexrd |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( abs ` ( G ` B ) ) e. RR* ) |
87 |
51
|
a1i |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 0 < 1 ) |
88 |
|
simplr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 1 < ( abs ` ( G ` B ) ) ) |
89 |
82 84 86 87 88
|
xrlttrd |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 0 < ( abs ` ( G ` B ) ) ) |
90 |
23
|
ffvelrnda |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( F ` u ) e. RR ) |
91 |
90
|
recnd |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( F ` u ) e. CC ) |
92 |
91
|
abscld |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( abs ` ( F ` u ) ) e. RR ) |
93 |
46
|
adantr |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. RR ) |
94 |
80 89 85 92 93
|
lemuldiv3d |
|- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( abs ` ( F ` u ) ) <_ ( sup ( ( abs " ( F " RR ) ) , RR , < ) / ( abs ` ( G ` B ) ) ) ) |
95 |
94
|
ralrimiva |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> A. u e. RR ( abs ` ( F ` u ) ) <_ ( sup ( ( abs " ( F " RR ) ) , RR , < ) / ( abs ` ( G ` B ) ) ) ) |
96 |
23 55 95
|
imo72b2lem2 |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) <_ ( sup ( ( abs " ( F " RR ) ) , RR , < ) / ( abs ` ( G ` B ) ) ) ) |
97 |
96 53 16 46 46
|
lemuldiv4d |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) x. ( abs ` ( G ` B ) ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) ) |
98 |
49 97
|
eqbrtrrd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( ( abs ` ( G ` B ) ) x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) ) |
99 |
6
|
adantr |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> E. x e. RR ( F ` x ) =/= 0 ) |
100 |
5
|
adantr |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 ) |
101 |
23 99 100
|
imo72b2lem1 |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 0 < sup ( ( abs " ( F " RR ) ) , RR , < ) ) |
102 |
98 101 46 16 46
|
lemuldiv3d |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) <_ ( sup ( ( abs " ( F " RR ) ) , RR , < ) / sup ( ( abs " ( F " RR ) ) , RR , < ) ) ) |
103 |
26 46
|
sseldd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. CC ) |
104 |
101
|
gt0ne0d |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) =/= 0 ) |
105 |
103 104
|
dividd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) / sup ( ( abs " ( F " RR ) ) , RR , < ) ) = 1 ) |
106 |
105
|
eqcomd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 1 = ( sup ( ( abs " ( F " RR ) ) , RR , < ) / sup ( ( abs " ( F " RR ) ) , RR , < ) ) ) |
107 |
102 106
|
breqtrrd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) <_ 1 ) |
108 |
16 17 107
|
lensymd |
|- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> -. 1 < ( abs ` ( G ` B ) ) ) |
109 |
11 108
|
pm2.65da |
|- ( ph -> -. 1 < ( abs ` ( G ` B ) ) ) |
110 |
9 10 109
|
nltled |
|- ( ph -> ( abs ` ( G ` B ) ) <_ 1 ) |